DOI QR코드

DOI QR Code

DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS

  • Received : 2012.10.19
  • Published : 2013.09.30

Abstract

Let R be a prime ring, I a nonzero ideal of R, $d$ a derivation of R, $m({\geq}1)$, $n({\geq}1)$ two fixed integers and $a{\in}R$. (i) If $a((d(x)y+xd(y)+d(y)x+yd(x))^n-(xy+yx))^m=0$ for all $x,y{\in}I$, then either $a=0$ or R is commutative; (ii) If $char(R){\neq}2$ and $a((d(x)y+xd(y)+d(y)x+yd(x))^n-(xy+yx)){\in}Z(R)$ for all $x,y{\in}I$, then either $a=0$ or R is commutative.

References

  1. N. Argac and H. G. Inceboz, Derivations of prime and semiprime rings, J. Korean Math. Soc. 46 (2009), no. 5, 997-1005. https://doi.org/10.4134/JKMS.2009.46.5.997
  2. M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math. 42 (2002), no. 1-2, 3-8. https://doi.org/10.1007/BF03323547
  3. L. Carini, V. De Filippis, and B. Dhara, Annihilators on co-commutators with generalized derivations on Lie ideals, Publ. Math. Debrecen 76 (2010), no. 3-4, 395-409.
  4. C. M. Chang and T. K. Lee, Annihilators of power values of derivations in prime rings, Comm. Algebra 26 (1998), no. 7, 2091-2113. https://doi.org/10.1080/00927879808826263
  5. C. L. Chuang, GPI's having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723-728. https://doi.org/10.1090/S0002-9939-1988-0947646-4
  6. V. De Filippis, Lie ideals and annihilator conditions on power values of commutators with derivation, Indian J. Pure Appl. Math. 32 (2001), no. 5, 649-656.
  7. B. Dhara, Power values of derivations with annihilator conditions on Lie ideals in prime rings, Comm. Algebra 37 (2009), no. 6, 2159-2167. https://doi.org/10.1080/00927870802226213
  8. B. Dhara, Annihilator condition on power values of derivations, Indian J. Pure Appl. Math. 42 (2011), no. 5, 357-369. https://doi.org/10.1007/s13226-011-0023-7
  9. B. Dhara, Left annihilators of power values of commutators with generalized derivations, Georgian Math. J. 19 (2012), no. 3, 441-448.
  10. T. S. Erickson, W. S. Martindale III, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49-63. https://doi.org/10.2140/pjm.1975.60.49
  11. I. N. Herstein, Topics in Ring Theory, Univ. of Chicago Press, Chicago, 1969.
  12. V. K. Kharchenko, Differantial identities of prime rings, Algebra i Logika 17 (1978), no. 2, 220-238.
  13. C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), no. 3, 731-734. https://doi.org/10.1090/S0002-9939-1993-1132851-9
  14. W. S.Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584. https://doi.org/10.1016/0021-8693(69)90029-5
  15. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100. https://doi.org/10.1090/S0002-9939-1957-0095863-0

Cited by

  1. A note on annihilator conditions in prime rings 2017, https://doi.org/10.1007/s12215-017-0305-y